Non-commutative computer algebra for polynomial algebras: Gröbner bases, applications and implementation (original) (raw)
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An introduction to commutative and noncommutative Gröbner bases
Theoretical Computer Science, 1994
Mora, T., An introduction to commutative and noncommutative Griibner bases, Theoretical Computer Science 134 (1994) 131-173. In 1965, Buchberger introduced the notion of Griibner bases for a polynomial ideal and an algorithm (Buchberger algorithm) for their computation; since the end of the seventies, Griibner bases have been an essential tool in the development of computational techniques for the symbolic solution of polynomial systems of equations and in the development of effective methods in Algebraic Geometry and Commutative Algebra; moreover, Grobner bases have been also generalized to free noncommutative algebra and to various noncommutative algebras, of interest in Differential Algebra (e.g. Weyl algebras, enveloping algebras of Lie algebras). The aim of this paper is to give an introduction, as elementary as I was able to make it, to both commutative and noncommutative algebras: Grobner bases are in a sense a finite model of an infinite linear Gauss-reduced basis of an ideal viewed as a vector space and Buchberger algorithm is the corresponding generalization of the Gaussian elimination algorithm. Moreover the paper contains a survey of some applications of Buchberger theory to noncommutative algebras; together with these results surveyed, this paper contains some minor new points: e.g. the "useless pair criteria" in the noncommutative case and the final result on the existence and "computability" of Griibner bases for two-sided ideals in any finitely presented algebra.
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13, 2013
Recently, the notion of "letterplace correspondence" between ideals in the free associative algebra K X and certain ideals in the so-called letterplace ring K[X | N] has evolved. We continue this research direction, started by La Scala and Levandovskyy, and present novel ideas, supported by the implementation, for effective computations with ideals in the free algebra by utilizing the generalized letterplace correspondance. In particular, we provide a direct algorithm to compute Gröbner bases of non-graded ideals. Surprizingly we realize its behavior as "homogenizing without a homogenization variable". Moreover, we develop new shift-invariant data structures for this family of algorithms and discuss about them.
Modular Techniques for Noncommutative Gröbner Bases
Mathematics in Computer Science
In this note, we extend modular techniques for computing Gröbner bases from the commutative setting to the vast class of noncommutative G-algebras. As in the commutative case, an effective verification test is only known to us in the graded case. In the general case, our algorithm is probabilistic in the sense that the resulting Gröbner basis can only be expected to generate the given ideal, with high probability. We have implemented our algorithm in the computer algebra system Singular and give timings to compare its performance with that of other instances of Buchberger's algorithm, testing examples from D-module theory as well as classical benchmark examples. A particular feature of the modular algorithm is that it allows parallel runs.
Symmetry, Integrability and Geometry: Methods and Applications, 2020
Let p be a polynomial in several non-commuting variables with coefficients in a field K of arbitrary characteristic. It has been conjectured that for any n, for p multilinear, the image of p evaluated on the set M n (K) of n by n matrices is either zero, or the set of scalar matrices, or the set sl n (K) of matrices of trace 0, or all of M n (K). This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for n = 2 in Section 2, some decisive results for n = 3 in Section 3, and partial information for n ≥ 3 in Section 4, also for non-multilinear polynomials. In addition we consider the case of K not algebraically closed, and polynomials evaluated on other finite dimensional simple algebras (in particular the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other researches, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of art, and in the other hand providing counterexamples showing the boundaries of generalizations.
Polynomial Algebras and their Applications
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different classes are constructed. Applications to quantum mechanical systems including supersymmetric quantum mechanics are discussed
Basic Module Theory over Non-commutative Rings with Computational Aspects of Operator Algebras
Lecture Notes in Computer Science, 2014
The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects. Non-commutative ring, Finitely presented module, Free resolution, Ore extension, Non-commutative factorization, Eigenring, Jacobson normal form, PBW ring, PBW algebra, Gröbner basis, Filtered ring, Gelfand-Kirillov dimension, grade number Partially supported by the Spanish Ministerio de Ciencia en Innovación and the European Union-grant MTM2010-20940-C02-01. The author wishes to thank Thomas Cluzeau, Viktor Levandovskyy, Georg Regensburger, and the anonymous referees for their comments that lead to improve this paper. JOSÉ GÓMEZ-TORRECILLAS 5.3. Computation of the Gelfand-Kirillov Dimension 41 6. Appendix on Computer Algebra Systems (by V. Levandovskyy) 44 6.1. Functionality of Systems for D[x; σ, δ] 45 6.2. Functionality of Systems for Multivariate Ore Algebras 45 6.3. Functionality of Systems for PBW Algebras 46 6.4. Further Systems 48 References 48
Letterplace ideals and non-commutative Gröbner bases
Journal of Symbolic Computation, 2009
In this paper we propose a 1-to-1 correspondence between graded two-sided ideals of the free associative algebra and some class of ideals of the algebra of polynomials, whose variables are doubleindexed commuting ones. We call these ideals the ''letterplace analogues'' of graded two-sided ideals. We study the behaviour of the generating sets of the ideals under this correspondence, and in particular that of the Gröbner bases. In this way, we obtain a new method for computing non-commutative homogeneous Gröbner bases via polynomials in commuting variables. Since the letterplace ideals are stable under the action of a monoid of endomorphisms of the polynomial algebra, the proposed algorithm results in an example of a Buchberger procedure ''reduced by symmetry''. Owing to the portability of our algorithm to any computer algebra system able to compute commutative Gröbner bases, we present an experimental implementation of our method in Singular. By means of a representative set of examples, we show finally that our implementation is competitive with computer algebra systems that provide non-commutative Gröbner bases from classical algorithms.